Generalized Valence Bond Wavefunctions Research Articles

Dynamical Electron Correlation and the Chemical Bond. IV. Covalent Bonds in A2 Molecules (A = N-As and F-Br).

In a series of recent papers, we investigated the effect of dynamical electron correlation on the potential energy curves and spectroscopic constants of several diatomic molecules, including the simple diatomic hydrides (AH) and the more complex diatomic fluorides (AF) and homonuclear diatomic molecules (A2) with A = B-F (AF) or A = C-F (A2), respectively. Our goal was to understand the dependence of the dynamical electron correlation energy, EDEC, on the internuclear distance, R, and quantify how dynamical electron correlation influences the spectroscopic constants (De, Re, and ωe) of these molecules. At large R, we found that the magnitude of EDEC(R) had a simple dependence on R, with EDEC(R) increasing nearly exponentially with decreasing R. However, as R continued to decrease, there were significant variations in EDEC(R). These variations led to differing changes in the predicted spectroscopic constants of the molecules. In many molecules, the changes in EDEC(R) could be correlated with changes in the underlying spin-coupled generalized valence bond wave function, either in the orbitals or the spin-coupling coefficients. In the current paper, we extend these studies to higher main group elements, comparing the effects of EDEC(R) on P2 and As2 versus N2, and on Cl2 and Br2 versus F2. We find that there are significant differences between the effects of dynamical electron correlation on the molecules in the first and subsequent rows of the periodic table.

Block-Correlated Coupled Cluster Theory with up to Four-Pair Correlation for Accurate Static Correlation of Strongly Correlated Systems.

A block-correlated coupled cluster method with up to four-pair correlation based on the generalized valence bond wave function (GVB-BCCC4) is first implemented, which offers an alternative method for electronic structure calculations of strongly correlated systems. We developed some techniques to derive a set of compact and cost-effective equations for GVB-BCCC4, which include the definition of n-block (n = 1-4) Hamiltonian matrices, the combination of excitation operators, and the definition of independent amplitudes. We then applied the GVB-BCCC4 method to investigate several potential energy surfaces of strongly correlated systems with singlet ground states. Our calculations demonstrate that the GVB-BCCC4 method can provide nearly exact static correlation energies as the density matrix renormalization group method (on the basis of the same GVB orbitals). This work highlights the significance of four-pair correlation in quantitative descriptions of static correlation energy for strongly correlated systems.

Equation-of-Motion Block-Correlated Coupled Cluster Method for Excited Electronic States of Strongly Correlated Systems.

An equation-of-motion block-correlated coupled cluster method based on the generalized valence bond wave function (EOM-GVB-BCCC) is proposed to describe low-lying excited states for strongly correlated systems. The EOM-GVB-BCCC2b method with up to two-pair correlation has been implemented and tested for a few strongly correlated systems. For a water hexamer with stretched O-H bonds, which is beyond the capability of the CASSCF method, EOM-GVB-BCCC2b provides very close results as the density matrix renormalization group (DMRG). For four conjugated diradical species with triplet ground states, we found that their vertical S-T gaps from EOM-GVB-BCCC2b are also quite consistent with the DMRG results. This new method is expected to be a promising theoretical tool for describing the low-lying excited states of strongly correlated systems with large active spaces.

Efficient Implementation of Block-Correlated Coupled Cluster Theory Based on the Generalized Valence Bond Reference for Strongly Correlated Systems

An optimized implementation of block-correlated coupled cluster theory based on the generalized valence bond wave function (GVB-BCCC) for the singlet ground state of strongly correlated systems is presented. The GVB-BCCC method with two-pair correlation (GVB-BCCC2b) or up to three-pair correlation (GVB-BCCC3b) will be the focus of this work. Three major techniques have been adopted to dramatically accelerate GVB-BCCC2b and GVB-BCCC3b calculations. First, the GVB-BCCC2b and GVB-BCCC3b codes are noticeably optimized by removing redundant calculations. Second, independent amplitudes are identified by constraining excited configurations to be pure singlet states and only independent amplitudes need to be solved. Third, an incremental updating scheme for the amplitudes in solving the GVB-BCCC equations is adopted. With these techniques, accurate GVB-BCCC3b calculations are now accessible for systems with relatively large active spaces (50 electrons in 50 orbitals) and GVB-BCCC2b calculations are affordable for systems with much larger active spaces. We have applied GVB-BCCC methods to investigate three typical kinds of systems: polyacenes, pentaprismane, and [Cu2O2]2+ isomers. For polyacenes, we demonstrate that GVB-BCCC3b can capture more than 94% of the total correlation energy even for 12-acene with 50 π electrons. For the potential energy curve of simultaneously stretching 15 C-C bonds in pentaprismane, our calculations show that the GVB-BCCC3b results are quite close to the results from the density matrix renormalization group (DMRG) over the whole range. For two dinuclear copper oxide isomers, their relative energy predicted by GVB-BCCC3b is also in good accord with the DMRG result. All calculations show that the inclusion of three-pair correlation in GVB-BCCC is critical for accurate descriptions of strongly correlated systems.

Dynamical electron correlation and the chemical bond. I. Covalent bonds in AH and AF (A = B-F).

Dynamical electron correlation has a major impact on the computed values of molecular properties and the energetics of molecular processes. This study focused on the effect of dynamical electron correlation on the spectroscopic constants (Re, ωe, De), and potential energy curves, ΔE(R), of the covalently bound AH and AF molecules, A = B-F. The changes in the spectroscopic constants (ΔRe, Δωe, ΔDe) caused by dynamical correlation are erratic and, at times, even surprising. These changes can be understood based on the dependence of the dynamical electron correlation energies of the AH and AF molecules as a function of the bond distance, i.e., ΔEDEC(R). At large R, the magnitude of ΔEDEC(R) increases nearly exponentially with decreasing R, but this increase slows as R continues to decrease and, in many cases, even reverses at very short R. The changes in ΔEDEC(R) in the region around Re were as unexpected as they were surprising, e.g., distinct minima and maxima were found in the curves of ΔEDEC(R) for the most polar molecules. The variations in ΔEDEC(R) for R ≲ Re are directly correlated with major changes in the electronic structure of the molecules as revealed by a detailed analysis of the spin-coupled generalized valence bond wave function. The results reported here indicate that we have much to learn about the nature of dynamical electron correlation and its effect on chemical bonds and molecular properties and processes.

Dispersion Interactions between Molecules in and out of Equilibrium Geometry: Visualization and Analysis.

The London dispersion interactions between systems undergoing bond breaking, twisting, or compression are not well studied due to the scarcity and the high computational cost of methods being able to describe both the dynamic correlation and the multireference character of the system. Recently developed methods based on the Generalized Valence Bond wave function, such as EERPA-GVB and SAPT(GVB) (SAPT = symmetry-adapted perturbation theory), allow one to accurately compute and analyze noncovalent interactions between multireference systems. Here, we augment this analysis by introducing a local indicator for dispersion interactions inspired by Mata and Wuttke’s Dispersion Interaction Density [J. Comput. Chem.2017, 38, 15−2327761924] applied on top of an EERPA-GVB computation. Using a few model systems, we show what insights into the nature and evolution of the dispersion interaction during bond breaking and twisting such an approach is able to offer. The new indicator can be used at a minimal cost additional to an EERPA-GVB computation and can be complemented by an energy decomposition employing the SAPT(GVB) method. We explain the physics behind the initial increase, followed by a decrease in the interaction of linear molecules upon bond stretching. Namely, the elongation of covalent bonds leads to the enhancement of attractive dispersion interactions. For even larger bond lengths, this effect is canceled by the increase of the repulsive exchange forces resulting in a suppression of the interaction and finally leading to repulsion between monomers.

Dressing of Vertices by Cumulants in Multi-Reference Coupled Cluster.

A new scheme is introduced in Multi-Reference (MR) Coupled Cluster (CC) based on the MR Generalized Normal Ordering (MRGNO) and the corresponding MR Generalized Wick Theorem (MRGWT) of Kutzelnigg and Mukherjee. The key element is the identification of a structure in MRGWT generated terms, facilitated by Goldstone diagram techniques. This allows for bundling the many terms of the MRGWT expansion and introduces a hierarchy in the equations that can be harnessed in devising approximations. One- and two-particle interaction vertices are found to be uniformly substituted for their counterpart dressed by density cumulants. This allows for a straightforward rewriting of the ordinary energy expression of CC with interaction dressed (id) one- and two-particle terms and reveals the presence of three- and higher-rank dressed interaction vertices too. Cumulants appearing out of dressed interaction vertices contribute to the amplitude equations and can be interpreted to have an amplitude dressing role. Dressing of one- and two-particle interaction vertices is the most straightforward to implement and does not hinder computational feasibility, provided that the reference function involves strictly limited active space sizes. The Generalized Valence Bond wave function, underlying pilot numerical tests, fulfills this criterion. Results on multiple bond breaking scenarios point to the need of stepping beyond one- and two-particle id. An extremely simple version of incorporating amplitude dressing in addition to one- and two-particle id is seen to cure the potential energy curves remarkably, stimulating further investigations along this line.

Calculation of dispersion interactions with the geminal-based ring Coupled Cluster Doubles method

The performance of the recently developed multi-reference extension of ring coupled cluster doubles is investigated for dispersion energy calculations, applied to the generalized valence bond wave function. The leading-order contribution to the dispersion energy is shown to have the correct asymptotic behaviour. Illustrative calculations on noble gas dimers are presented.

Automatic Construction of the Initial Orbitals for Efficient Generalized Valence Bond Calculations of Large Systems.

We propose an efficient general strategy for generating initial orbitals for generalized valence bond (GVB) calculations which makes routine black-box GVB calculations on large systems feasible. Two schemes are proposed, depending on whether the restricted Hartree-Fock (RHF) wave function is stable (scheme I) or not (scheme II). In both schemes, the first step is the construction of active occupied orbitals and active virtual orbitals. In scheme I, active occupied orbitals are composed of the valence orbitals (the inner core orbitals are excluded), and the active virtual orbitals are obtained from the original virtual space by requiring its maximum overlap with the virtual orbital space of the same system at a minimal basis set. In scheme II, active occupied orbitals and active virtual orbitals are obtained from the set of unrestricted natural orbitals (UNOs), which are transformed from two sets of unrestricted HF spatial orbitals. In the next step, the active occupied orbitals and active virtual ones are separately transformed to localized orbitals. Localized occupied and virtual orbital pairs are formed using the Kuhn-Munkres (KM) algorithm and are used as the initial guess for the GVB orbitals. The optimized GVB wave function is obtained using the second-order self-consistent-field algorithm in the GAMESS program. With this procedure, GVB energies have been obtained for the lowest singlet and triplet states of polyacenes (up to decacene with 96 pairs) and the singlet ground state of two di-copper-oxygen-ammonia complexes. We have also calculated the singlet-triplet gaps for some polyacenes and the relative energy between two di-copper-oxygen-ammonia complexes with the block-correlated second-order perturbation theory based on the GVB reference.

Insights into the Electronic Structure of Molecules from Generalized Valence Bond Theory

In this article we describe the unique insights into the electronic structure of molecules provided by generalized valence bond (GVB) theory. We consider selected prototypical hydrocarbons as well as a number of hypervalent molecules and a set of first- and second-row valence isoelectronic species. The GVB wave function is obtained by variationally optimizing the orbitals and spin coupling in the valence bond wave function. The GVB wave function is a generalization of the Hartree-Fock (HF) wave function, lifting the double occupancy restriction on a subset of the HF orbitals as well as the associated orthogonality and spin coupling constraints. The GVB wave function includes a major fraction (if not all) of the nondynamical correlation energy of a molecule. Because of this, GVB theory properly describes bond formation and can answer one of the most compelling questions in chemistry: How are atoms changed by molecular formation? We show that GVB theory provides a unified description of the nature of the bonding in all of the above molecular species as well as contributing new insights into the well-known, but poorly understood, first-row anomaly.

Insights into the electronic structure of disulfur tetrafluoride isomers from generalized valence bond theory.

Sulfur and fluorine can participate in a variety of bonding motifs, lending significant diversity to their chemistry. Prior work has identified three distinct minima for disulfur tetrafluoride (S2F4) compounds: two FSSF3 isomers and one SSF4 species. We used a combination of sophisticated explicitly correlated coupled cluster calculations and generalized valence bond (GVB) theory to characterize the electronic structure of these species as well as additional stationary points on the potential energy surface with F2SSF2 connectivity. On the singlet surface, the two stationary points considered in this work with an F2SSF2 structure are first- or second-order saddle points and not minima. However, on the triplet surface, we discovered a novel C2 symmetric F2SSF2 minimum that was anticipated from the structure of an excited state ((3)B1) of SF2. Analysis using the GVB wave function in conjunction with the recoupled pair bonding model developed by our group provides a straightforward explanation of the bonding in all of the S2F4 structures considered here. In addition, the model predicted the existence of the F2SSF2((3)B) minimum.

Insights into the Perplexing Nature of the Bonding in C2 from Generalized Valence Bond Calculations.

Diatomic carbon, C2, has been variously described as having a double, triple, or quadruple bond. In this article, we report full generalized valence bond (GVB) calculations on C2. The GVB wave function-more accurate than the Hartree-Fock wave function and easier to interpret than traditional multiconfiguration wave functions-is well-suited for characterizing the bonding in C2. The GVB calculations show that the electronic wave function of C2 is not well described by a product of singlet-coupled, shared electron pairs (perfect pairing), which is the theoretical basis for covalent chemical bonds. Rather, C2 is best described as having a traditional covalent σ bond with the electrons in the remaining orbitals of the two carbon atoms antiferromagnetically coupled. However, even this description is incomplete as the perfect pairing spin function also makes a significant contribution to the full GVB wave function. The complicated structure of the wave function of C2 is the source of the uncertainty about the nature of the bonding in this molecule.

On the electronic structure of the diazomethane molecule

The electronic structure and chemical bonding in the diazomethane molecule are investigated using full-valence generalized valence bond (GVB) methods. We point out that an ab initio-based bonding model must correspond directly to a wave function that yields at least qualitatively corrects values for the structural parameters of a molecule, that is, molecular geometry, vibrational frequencies, and dipole moment. However, in the case of diazomethane, when trying to emulate the bonding models proposed in the literature through full-valence GVB wave functions, we found out that all of them are directly associated with optimized molecular geometries that are saddle points in the molecular potential energy surface. This spurious behavior is corrected by a multiconfiguration–self-consistent field (MCSCF) wave function that incorporates an enlarged “pi-like” active space enabling a complete active space self-consistent field (CASSCF) block, with more active orbitals than electrons, together with a “sigma-like” generalized valence bond with restricted configuration interaction (GVB-RCI) block. With this wave function, we are able to generate the best calculated set of harmonic frequencies to date for the diazomethane molecule. The physical effects that are important for the correct description of its electronic and vibrational structure are then discussed using a series of MCSCF wave functions. This result leads to a decomposition of the electronic wave function into diabatic GVB-RCI chemical structures along the CH2 wagging mode illustrating the necessity to understand the chemical bonding in this molecule as a superposition of bonding patterns. Some structural properties of diazomethane and diazocompounds are then successfully analyzed using our model.

Chemical Bonding and the Equilibrium Composition of Grignard Reagents in Ethereal Solutions

A thorough analysis of the electronic structure and thermodynamic aspects of Grignard reagents and its associated equilibrium composition in ethereal solutions is performed. Considering methylmagnesium halides containing fluorine, chlorine, and bromine, we studied the neutral, charged, and radical species associated with their chemical equilibrium in solution. The ethereal solvents considered, tetrahydrofuran (THF) and ethyl ether (Et(2)O), were modeled using the polarizable continuum model (PCM) and also by explicit coordination to the Mg atoms in a cluster. The chemical bonding of the species that constitute the Grignard reagent is analyzed in detail with generalized valence bond (GVB) wave functions. Equilibrium constants were calculated with the DFT/M06 functional and GVB wave functions, yielding similar results. According to our calculations and existing kinetic and electrochemical evidence, the species R(•), R(-), (•)MgX, and RMgX(2)(-) must be present in low concentration in the equilibrium. We conclude that depending on the halogen, a different route must be followed to produce the relevant equilibrium species in each case. Chloride and bromide must preferably follow a "radical-based" pathway, and fluoride must follow a "carbanionic-based" pathway. These different mechanisms are contrasted against the available experimental results and are proven to be consistent with the existing thermodynamic data on the Grignard reagent equilibria.

The electronic structure of the F2, Cl2, Br2 molecules: the description of charge-shift bonding within the generalized valence bond ansatz

In this paper, we present an alternative picture for the electronic structure of dihalogen molecules and for the physical origin of the “charge-shift bonding” effect. Absolute energies, binding energies, quadrupole moments and harmonic frequencies are determined for a hierarchy of methods from Hartree-Fock (HF), many forms of generalized valence bond (GVB) wavefunctions to Multi-Reference-MP2. All valence electron pairs are explicitly correlated in the GVB wavefunctions. It is shown that HF charge densities for the fluorine molecule are extremely inaccurate. This fact causes the HF canonical orbital basis for this molecule to be inadequate in low order correlation treatments in spite of the fact that there are no “near degeneracies” at the equilibrium distance. The accurate description charge fluctuation lone pair repulsions are essential for a proper assessment of the fluorine molecule binding energy, bond distance and harmonic frequency. These properties are well described by lifting the perfect-pairing restriction in a full-valence orbital optimized GVB-RCI (restricted-configuration-interaction) wavefunction. The accurate calculation of electron–electron cusps is of lesser importance in the description of the electronic structure of the fluorine molecule than usually considered. An analysis of the lone pair GVB natural orbitals provides a clear-cut understanding on the differences between fluorine and the other dihalogen molecules. Within our model, we conclude that among the dihalogen molecules the charge-shift bonding concept is meaningful only for the fluorine molecule.

On the nodal structure of single-particle approximation based atomic wave functions

The nodal structures of atomic wave functions based on a product of spatial orbitals, namely, restricted, unrestricted, and generalized valence bond wave functions, are shown to be equivalent. This result is verified by fixed node-diffusion Monte Carlo simulations for atoms up to Ne. Also for a molecular system, Li(2) at the equilibrium geometry, a multideterminantal generalized valence bond wave function does not improve the nodal surfaces of a restricted Hartree-Fock wave function.

Effects of J-gate potential and uniform electric field on a coupled donor pair in Si for quantum computing

We present theoretical studies of a coupled-donor pair in Si via an unrestricted Hartree-Fock method with a generalized valence bond wave function. Polarization properties and exchange coupling for a phosphorous donor pair in silicon under a J-gate potential (modeled by a parabolic well) and a uniform electric field (either parallel or perpendicular to the interdonor axis) are examined. The energies and charge distributions of the lowest-lying singlet and triplet states as functions of the J-gate potential and uniform electric field for various donor separations are analyzed. Implications for Si:P electron-spin-based quantum computer architecture are discussed.

Theoretical description of the low-lying valence states of the MgC molecule

Restricted Hartree–Fock (RHF) and generalized valence bond (GVB) calculations, plus configuration interaction (CI), have been performed for four electronic valence states [3Σ− (1) 5Σ− (2) 3Π, (3) 3Π (4)] of the MgC molecule. Their energies have been calculated, in the range from 1.5 to 15.0 Å. From the potential curves spectroscopic constants have been obtained and the respective products of dissociation well characterized. The GVB wavefunctions allowed us to understand the bonding details and to identify the stabilization effects for the different states of the molecule. The 3Σ− state was found to be the ground state, in agreement with previous calculations, with the 5Σ− lying very close above. © 1995 John Wiley & Sons, Inc.

A generalized direct inversion in the iterative subspace approach for generalized valence bond wave functions

We present a greatly improved method for converging generalized valence bond (GVB) self-consistent wave functions. This method starts with the direct inversion in the interative subspace (DIIS) ideas of Pulay. Previously implemented DIIS methods were limited to special cases: closed-shell Hartree–Fock (HF), restricted open-shell HF, or a single pair GVB wave function. Here we extend this method to general wave functions including arbitrary numbers of closed-shell, restricted open-shell, and GVB orbitals (including second-order orbital mixing terms). The efficacy of GVB-DIIS is illustrated by applying it to several cases (including GVB wave functions with up to ten pairs) and comparing with other standard methods.

A generalized valence bond representation of complete-active-space self-consistent-field (CASSCF) wave functions

A novel generalized valence bond wave function expressed in terms of local nonorthogonal orbitals is shown to attain nearly all of the correlation energy of a corresponding complete-active-space self-consistent-field (CASSCF) wave function. This new wave function allows for the local orbitals to variationally respond to different spin couplings. The result is a very compact description of correlation effects contained in complex CASSCF expansions. The CASSCF expansion is also shown to contain a description of resonance among nonorthogonal valence bond structures.